Establishing Paternity: How Can an Adult Correct a Birth Certificate That Lists the Wrong Name for the Father

My question involves paternity law for the State of New Jersey.

For reasons unknown, my mother decided to put a fake name for my father on my birth certificate instead of my biological father. This is a completely fake name and not the name of another man. When asked why, I do not receive an answer. Now I am 28 years old and I need a certified copy of my birth certificate. My mother does not remember the fake name she put on the birth certificate. I have been denied a request for a copy of my birth certificate as the information I provide does not match and I assume it's the father's name on the record.

Both of my biological parents are still together and have been since before I was born. They have not gotten married. Is there any action I can take towards getting a copy of the birth certificate, such as changing the original documentation or proving my biological father through a DNA test? What legal trouble can my mother face for writing a completely fake name?

Establishing Paternity: How Can an Adult Correct a Birth Certificate That Lists the Wrong Name for the Father